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Maths-

General

Easy

Question

If n is even, then in the expansion of $open parentheses 1 plus fraction numerator x to the power of 2 end exponent over denominator 2 factorial end fraction plus fraction numerator x to the power of 4 end exponent over denominator 4 factorial end fraction plus midline horizontal ellipsis close parentheses to the power of 2 end exponent$ , the coefficient of $x to the power of n end exponent text is end text$

$\frac{2^{n}}{n!}$
$\frac{2^{n} - 2}{n!}$
$\frac{2^{n - 1} - 1}{n!}$
$\frac{2^{n - 1}}{n!}$

The correct answer is: $fraction numerator 2 to the power of n minus 1 end exponent over denominator n factorial end fraction$

$open parentheses 1 plus fraction numerator x to the power of 2 end exponent over denominator 2 factorial end fraction plus fraction numerator x to the power of 4 end exponent over denominator 4 factorial end fraction plus midline horizontal ellipsis close parentheses to the power of 2 end exponent equals open parentheses fraction numerator e to the power of x end exponent plus e to the power of negative x end exponent over denominator 2 end fraction close parentheses to the power of 2 end exponent equals fraction numerator 1 over denominator 4 end fraction open parentheses e to the power of 2 x end exponent plus e to the power of negative 2 x end exponent plus 2 close parentheses$
$equals fraction numerator 1 over denominator 4 end fraction open curly brackets 2 open parentheses 1 plus fraction numerator left parenthesis 2 x right parenthesis to the power of 2 end exponent over denominator 2 factorial end fraction plus fraction numerator left parenthesis 2 x right parenthesis to the power of 4 end exponent over denominator 4 factorial end fraction plus midline horizontal ellipsis close parentheses plus 2 close curly brackets$
$text Therefore, the coefficient of end text x to the power of n end exponent left parenthesis n text even end text right parenthesis$
$fraction numerator 1 over denominator 2 end fraction open curly brackets fraction numerator 2 to the power of n end exponent over denominator n factorial end fraction close curly brackets equals fraction numerator 2 to the power of n minus 1 end exponent over denominator n factorial end fraction$

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$text If end text 0 less than alpha less than fraction numerator pi over denominator 2 end fraction text and end text f left parenthesis alpha right parenthesis equals pi to the power of sec to the power of 2 end exponent invisible function application alpha end exponent c o s to the power of 2 end exponent invisible function application alpha plus pi to the power of cosec to the power of 3 end exponent invisible function application alpha end exponent s i n to the power of 2 end exponent invisible function application alpha comma text end text$ then

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